Approximately Additive and Approximately Linear Mappings
D. H. Hyers, George Isac, Themistocles M. Rassias · Birkhäuser Boston eBooks · 1998
We have mentioned the concept of stability and given an example for the Cauchy functional equation (1.1). A definition of stability in the case of homomorphisms between metric groups was suggested by a problem posed by S.M. Ulam in 1940 (see Ulam (1960), p. 64). Given a group ( G 1 , •), a metric group ( G 2 ,*) with metric d and a positive number η , suppose that there exists a positive number ε= ε(η) such that, if d ( f ( x • y ), f ( x ) * f ( y )) < ε for some f : G 1 → G 2 and all x and y in G 1 , then a homomorphism h: G 1 → G 2 exists with d ( f ( x ), h ( x )) < η for all x in G 1 . In this case, the equation of homomorphism h ( x • y ) = h ( x ) * h ( y ) is called stable . Theorem 1.1 with G 1 = E 1 , G 2 = E 2 and with addition as the group operation in each case shows that Cauchy’s equation is stable by this definition with η= ε .